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\begin{document}
\title{GenSAT: A Navigational Approach}
\author{Yury Smirnov{1} and Manuela M. Veloso{2}}
\institute{
School of Computer Science \\
Carnegie Mellon University \\
Pittsburgh, PA 15213-3891 \\
\{smir, mmv\}@cs.cmu.edu}
%\date{\vspace*{-3cm}}
\maketitle
\begin{abstract}
GenSAT is a family of local hill-climbing procedures for solving
propositional satisfiability problems. We restate it as a navigational
search process performed on an $N$-dimensional cube by a fictitious
agent with limited lookahead. Several members of the GenSAT family have
been introduced whose efficiency varies from the best in average for
randomly generated problems to a complete failure on the realistic,
specially constrained problems, hence raising the interesting
question of understanding the essence of their different performance.
In this paper, we show how we use our navigational approach to
investigate this issue. We introduce new algorithms that sharply focus
on specific combinations of properties of efficient GenSAT variants, and
which help to identify the relevance of the algorithm features to the
efficiency of local search. In particular, we argue for the reasons
of higher effectiveness of HSAT compared to the original GSAT. We
also derive fast approximating procedures based on variable weights
that can provide good switching points for a mixed search policy. Our
conclusions are validated by empirical evidence obtained from the
application of several GenSAT variants to random 3SAT problem instances
and to simple navigational problems.
\end{abstract}
\section{Introduction}
Recently an alphabetical mix of variants of GSAT \cite{gu:92,selman:92}
has
attracted a lot of attention from Artificial Intelligence (AI)
researchers:
TSAT, CSAT, DSAT, HSAT \cite{gent:93,gent:95}, WSAT \cite{selman:94},
WGSAT,
UGSAT \cite{frank:96} just to name few. All these local hill-climbing
procedures
are members of the GenSAT family. Propositional satisfiability (SAT) is
the fundamental problem of the class of NP-hard problems, which is
believed not
to admit solutions that are always polynomial on the size of the
problems.
Many practical AI problems have been directly encoded or reduced to SAT.
GenSAT (see Table~\ref{gensat}) is a family of hill-climbing procedures
that
are capable of finding satisfiable assignments for some large-scale
problems
that cannot be attacked by conventional resolution-based methods.
\begin{table}[bht]
\hrule
\begin{center}
\begin{minipage}[t]{12cm}
%\renewcommand{\baselinestretch}{1}
{\small {\normalsize
\begin{tabbing}
~~~\=~~~\=~~~\=~~~\=~~~\= \kill
{\bf procedure}\ GenSAT ($\Sigma$) \\
\> {\bf for}\ i:=1 {\bf to}\ Max\_Tries \\
\> \> T:= $initial(\Sigma)$ \\
\> \> {\bf for}\ j:=1 {\bf to}\ Max\_Flips \\
\> \> \> {\bf if}\ T satisfies $\Sigma$ {\bf then return}\ T \\
\> \> \> {\bf else {\em poss-flips}} := $hill$-$climb(\Sigma,T)$ \\
\> \> \> \> \> ; compute best local neighbors of $T$ \\
\> \> \> \> V := $pick(${\bf {\em poss-flips}}$)$ ; pick a variable \\
\> \> \> \> T := T with V's truth assignment inverted \\
\> \> {\bf end} \\
\> {\bf end} \\
{\bf return}\ ``no satisfying assignment found'' \\
\end{tabbing}
}}
\end{minipage}
\end{center}
\vspace*{-0.45cm}
\hrule
\caption{The GenSAT Procedure.}
\label{gensat}
\end{table}
GSAT \cite{gu:92,selman:92} is an instance of GenSAT in which {\em
initial}
(see Table~1) generates a random truth assignment, {\em hill-climb}
returns all those
variables whose flips\footnote{Flip is a change of the current value of a
variable to
the opposite value.} give the greatest increase in the number of
satisfied
clauses and {\em pick} chooses one of these variables at random
\cite{gent:93}.
Previous work on the behavior of GSAT and similar hill-climbing
procedures \cite{gent:93} identified two distinct search phases and
suggested
possible improvements for GenSAT variants. HSAT is a specific variant of
GenSAT,
which uses a queue to control the selection of variables to
flip\footnote{See Section3
for the definition of HSAT.}. Several research efforts has attempted to
analyze
the dominance of HSAT compared with the original GSAT for randomly
generated
problem instances. We have developed a navigational search framework that
mimics the behavior of GenSAT. This navigational approach allows us to
re-analyze
the reasons of higher effectiveness of HSAT and other hill-climbing
procedures by relating
it to the number of equally good choices. This navigational approach also
suggests strong approximating SAT procedures that can be applied
efficiently to practical problems. An approximation approach can be
applied
to both ``easy'' and ``hard'' practical problems, in the former case it
will
likely to produce a satisfiable assignment, whereas in the latter case it
will
quickly find an approximate solution. For a standard testbed of randomly
generated 3SAT problems, the transition phase between ``easy'' and
``hard''
problem instances corresponds to the ratio value of $4.3$ between the
number of
clauses $L$ to the number of variables
$N$~\cite{mitchell:92,crawford:93}.
Figure~\ref{transit_phase} demonstrates the probability of generating a
satisfying
assignment for random 3SAT problems depending on the $L/N$-ratio.
\putfig{transit_phase}{The transition phase for random 3SAT
problems.}{width=0.4\textwidth}
%Since the ratio between the number of clauses
%$L$ to the number of variables $N$ in practical instances of 3SAT is not
always
%equal to 4.3 (which is recognized as the hardest for resolving 3SAT
problem
%instances\cite{mitchell:92,crawford:93}), an approximation approach can
be
%applied. It solves efficiently ``easy'' problems with $L\leq 4.3N$, or
%finds quickly an approximate solution for ``hard'' problems with
%$L\geq 4.3N$. When the ratio of the number of clauses to the number of
%variables approaches $4.3$, the procent of the satisfiable 3SAT formulas
%experiences the transition phase.
An approximate solution can be utilized in problems with time-critical or
dynamically changing domains. Interestingly, we found that it also
provides
a good starting point for a different search policy, i.e. serves as a
switching
point between distinct search policies within the same procedure. Such an
approach can be utilized beneficially in multi-processor/multi-agent
problem
settings.
Our experiments with randomly generated 3SAT problem instances and
realistic
navigational problems confirmed the results of our analysis.
%\puttable{gsat}{The GSAT Procedure.}{width=0.45\textwidth}
\section{GenSAT as an Agent-Centered Search}
State spaces for boolean satisfiability problems can be represented as
$N$-dimensional cubes, where $N$ is the number of variables. We view GSAT
and
similar hill-climbing procedures as performing search on these highdimensional
cubes by moving a fictitious agent with limited lookahead. For efficiency
reasons, the majority of GSAT-like procedures limit the lookahead of the
agent
to the neighbors of its current state, i.e., to those vertices of the
cube
that are one step far from the current vertex. An edge of the cube that
links two neighboring vertices within the same face of the cube,
corresponds
to the flip of a variable. Thus, we reduced the behavior of GSAT to
agent-centered search on a high-dimensional cube. Recall, in agentcentered
search the search space is explored incrementally by an agent with
limited
lookahead. Throughout the paper we refer to this navigational version of
GenSAT
as to NavGSAT.
The worst-case complexity of both informed and uninformed agent-centered
search is of the order of the number of vertices, i.e. $O(2^N)$.
Moreover, unlike
classical AI search where A* is an optimal informed algorithm for
an arbitrary admissible heuristic, there are no optimal algorithms for
agent-centered search problems\cite{smirnov:96}. Furthermore, even a
consistent, admissible heuristic can become misleading, and an efficient
informed agent-centered search algorithm can demonstrate worse
performance
than the uninformed (zero heuristic) version of the same
algorithm \cite{koenig:96}.
From the algorithmic point of view, the behavior of LRTA*~\cite{korf:90},
one
of the most efficient agent-centered search methods, is close to
NavGSAT's behavior.
Both methods look for the most promising vertex among neighbors of the
current
vertex. In addition to selecting a neighbor with the best heuristic
value,
LRTA* also updates the heuristic value of the current vertex (see
Table~\ref{lrta}).
The efficiency of LRTA* depends on how closely the heuristic function
represents the real distance \cite{smirnov:96}. The vast majority of
GSAT-like
procedures use the number of unsatisfied (or satisfied) clauses as the
guiding
heuristic. In general, this heuristic is neither consistent, nor
admissible.
However, for the most intricate random instances of SAT problems with
$L=O(N)$, this heuristic is an $O(N)$ approximation of the real distance.
Therefore, $\epsilon$-search \cite{ishida:96}, a modification of LRTA*
that uses approximations of admissible heuristics, applies to SAT
problems.
\comment{
\puttableb{lrta}{Learning Real-Time Algorithm (LRTA*). LRTA* also looks
for the most
promising vertex among neighbors of the current
vertex.}{width=0.45\textwidth}
}
\begin{table}[tbh]
\hrule
\begin{center}
\begin{minipage}[t]{12cm}
\renewcommand{\baselinestretch}{1}
{\normalsize {\small
\begin{tabbing}
~~~\=~~~\=~~~\= \kill
{\bf procedure}\ LRTA*$(V,E)$ \\
\> Initially, $F(v) := h(v)$ for all $v\in V$. \\
\> LRTA* starts at vertex $v_{start}$:\\ \\
\> \> 1. $v :=$ the current vertex. \\
\> \> 2. If $v\in Goal$, then STOP successfully. \\
\> \> 3. $e := argmin_e F(neighbor(v,e))$. \\
\> \> 4. $F(v) := max(F(v), 1+F(neighbor(v,e)))$. \\
\> \> 5. Traverse edge $e$, update $v := neighbor(v,e)$. \\
\> \> 6. Go to 2. \\
\end{tabbing}
}}
\end{minipage}
\end{center}
\vspace*{-0.45cm}
\hrule
\caption{Learning Real-Time Algorithm (LRTA*). LRTA* also looks for the
most
promising vertex among neighbors of the current vertex.}
\label{lrta}
\end{table}
\begin{lemma} \label{th1}
After repeated problem-solving trials of a soluble propositional
satisfiability problem with $N$ variables and $O(N)$ clauses, the length
of
the solution of $\epsilon$-search converges to $O(N^2)$.
\end{lemma}
{\bf Proof:}\ After repeated problem-solving trials the length of a
solution of
$\epsilon$-search converges to the length of the optimal path multiplied
by
$(1+\epsilon)$ \cite{ishida:96}. On one hand, the length of the optimal
path
for a soluble propositional satisfiability problem is $O(N)$. On the
other
hand, for problems with $L = O(N)$ approximating factor $\epsilon$ is
also
$O(N)$. These two facts imply $O(N^2)$ complexity of the final solution
after an unknown number of repeated trials.\sqed
Even though the length of a solution of $\epsilon$-search converges to
$O(N^2)$
for soluble problem instances, several initial trials can have
exponential
length. Thus, this approach can be applied only in special circumstances:
One is provided possibly exponential memory and possibly exponential time
for pre-processing to re-balance the heuristic values, then the
complexity of
solving of the pre-processed problem is $O(N^2)$. Since this scenario is
not
always what AI researchers keep in mind when applying GenSAT, we do not
consider
$\epsilon$-search as a general navigational equivalent of GenSAT.
However, in
Section~3 we show that one (first) run of $\epsilon$-search coincides
completely
with the run of HSAT for the majority of soluble SAT problem instances.
Thus, the question of the efficiency of GSAT and similar procedures is
reduced
to the domain-heuristics relations that guide agent-centered search on an
$N$-dimensional cube. Recent works on changing the usual static heuristic
-the number of unsatisfied (satisfied) clauses -- to the dynamic weighted
sums
\cite{frank:96} produced another promising sub-family of GenSAT
procedures. Our
experiments showed that the ``quality'' of the usual heuristic varies
greatly in
different regions of the $N$-dimensional cube, and as the ratio of $L$ to
$N$
grows, this heuristic becomes misleading in some regions of the problem's
domain. These experiments identified the need to introduce novel
heuristics and better analysis of the existing ones.
\section{New Corners or Branching Factor?}
We conducted a series of experiments with the $\epsilon$-search version
of LRTA* and the number of unsatisfied clauses as the heuristic values
for each
vertex (corner) of the $N$-dimensional cube. We found that the
combination of a
highly connected $N$-dimensional cube and such prior knowledge forces
an agent to avoid vertices with updated (increased in step~4) heuristic
values.
Exactly the same effect has been achieved by HSAT - a variant of GenSAT.
In HSAT flipped variables form a queue, and this queue is used in $pick$
to break ties in favor of variables flipped earlier until the satisfying
assignment is found or the amount of flips has reached the pre-set limit
of $Max\_Flips$. Thus, $\epsilon$-search is a navigational analogue of
HSAT
for soluble problem instances.
Previous research identified two phases of GenSAT procedures: steady
hill-climbing and plateau phases \cite{gent:93}. During the plateau phase
these procedures perform series of sideway flips keeping the number of
satisfied clauses on the same level. The reduction of the number of such
flips,
i.e. cutting down the length of the plateau, has been identified as the
main
concern of such procedures. Due to high connectivity of the problem
domain and
the abundance of equally good choices during the plateau phase, neither
HSAT
nor $\epsilon$-search re-visit already explored vertices (corners) of the
cube for
large-scale problems. This property of HSAT has been stated as
the reason of its performance advantage for randomly generated problems
in comparison with GSAT \cite{gent:95}.
To re-evaluate the importance of visiting new corners of the $N$dimensional cube,
we introduced another hill-climbing procedure, that differs from GSAT
only in
keeping track of all visited vertices and Never Re-visiting them again,
NRGSAT.
On all randomly generated 3SAT problems, NRGSAT's performance in terms
of flips was identical to GSAT's one. Practically, NRGSAT ran much
slower,
because it needs to maintain a list of visited vertices and check it
before
every flip. Based on this experiment, we were able to conclude that
exploring
new corners of the cube is not that important. This increased our
interest in
studying further reasons for the performance advantage of HSAT over GSAT.
We focused our attention on {\em poss-flips} -- the number of
equally good flips between which GSAT randomly picks \cite{gent:93a}, or,
alternatively, the branching factor of GSAT search during the plateau
phase.
We noticed that on earlier stages of the plateau phase both GSAT
and NRGSAT tend to increase {\em poss-flips}, whereas HSAT randomly
oscillates
{\em poss-flips} around a certain (lower) level. To confirm the
importance of
{\em poss-flips}, we introduced {\em variable weights}\footnote{Weight of
each
variable is the number of times this
variable has been flipped from the beginning of the search procedure.
Each flip of a variable increases its weight by one.} as a second
heuristic to break ties during the plateau phase of NavGSAT.
NavGSAT monitors the number of flips performed
for each variable and among all equally good flips in terms of the number
of
unsatisfied clauses, NavGSAT picks a variable that was flipped the least
number of
times. In case of second-order ties, they can be broken either randomly,
fair -NavRGSAT -- or deterministically, unfair, according to a fixed order -NavFGSAT.
\begin{table}[bht]
\begin{center}
\begin{minipage}[t]{10cm}
\renewcommand{\baselinestretch}{1}
{\normalsize {\small
\begin{tabular}{|c|c|c|c|c|} \hline
Problem & Algorithm & Mean & Median & St.Dev. \\ \hline
{\bf 100} & GSAT & 12,869 & 5326 & 9515 \\
{\bf vars,} & HSAT & 2631 & 1273 & 1175 \\
{\bf 430} & NavFGSAT & 3558 & 2021 & 1183 \\
{\bf clauses} & NavRGSAT & 3077 & 1743 & 1219 \\ \hline
{\bf 1000} & GSAT & 4569 & 2847 & 1863 \\
{\bf vars,} & HSAT & 1602 & 1387 & 334 \\
{\bf 3000} & NavFGSAT & 1475 & 1219 & 619 \\
{\bf clauses} & NavRGSAT & 1649 & 1362 & 675 \\ \hline
{\bf 1000} & GSAT & 7562 & 4026 & 3515 \\
{\bf vars,} & HSAT & 3750 & 2573 & 1042 \\
{\bf 3650} & NavFGSAT & 3928 & 2908 & 1183 \\
{\bf clauses} & NavRGSAT & 4103 & 3061 & 1376 \\ \hline
\end{tabular}
}}
\end{minipage}
\end{center}
\caption{Comparison of number of flips for GSAT, HSAT, NavRGSAT and
NavFGSAT.}
\label{table}
\end{table}
Both NavRGSAT and NavFGSAT allow to flip back the just flipped variable.
Moreover, the latter procedure often forces to do so due to the fixed
order of
variables. However, the performance of both NavRGSAT and NavFGSAT is very
close
to HSAT's performance. Table~\ref{table} presents median, mean and
standard
deviation of GSAT, HSAT, NavRGSAT and NavFGSAT for
randomly generated 3SAT problems with 100 and 1000 variables and
different ratios $L$
to $N$. We investigated problems of this big size, because they
represents the
threshold between satisfiability problems that accept solutions by
conventional
resolution methods, for example Davis-Putnam procedure, and ones that can
be
solved by GenSAT hill-climbing procedures.
In the beginning of the plateau phase both NavGSAT methods behave
similarly to
HSAT: Variables flipped earlier are considered last when NavGSAT is
looking
for the next variable to flip. As more variables gain weight,
NavGSAT methods' behavior deviates from HSAT. Both methods can be
perceived as an approximation of HSAT.
We identified that a larger number of {\em poss-flips} is the main reason
why
GSAT loses to HSAT and NavGSAT on earlier stages of the plateau phase. As
the
number of unsatisfied clauses degrades, there are less choices for
equally
good flips for GSAT, and the increase of {\em poss-flips} is less
visible.
During earlier sideway flips GSAT picks equally good variables randomly,
this
type of selection leads to the vertices of the cube with bigger
{\em poss-flips}, where GSAT tends to be ``cornered'' for a while.
Figure~\ref{tie-fig} presents average amounts of {\em poss-flips} with
the
95\%-confidence intervals. The {\em poss-flips} were summed up for each
out of
four hill-climbing procedures for every step in the beginning of the
plateau phases
(from $0.25N$ to $N$) for a range of problem sizes. Since the number of
variables
and the interval of measuring grow linearly on $N$, we present sums of
{\em poss-flips} scaled down by $N^2$. As it follows from
Figure~\ref{tie-fig}, the original GSAT consistently outnumbers all other
three procedures during that phase, although its confidence intervals
overlap with NavRGSAT and NavFGSAT's confidence intervals.
%\putfig{tie-fig}{Comparison of {\em Poss-Flips} for GSAT, HSAT, NavRGSAT
and NavFGSAT.}{width=0.45\textwidth}
\begin{figure}[t] \centering \
\psfig{figure=figs/tie-fig.eps,width=0.3\textwidth}
\psfig{figure=figs/tie-fig-scaled.eps,width=0.3\textwidth}
\psfig{figure=figs/tie-fig-nfgsat.eps,width=0.3\textwidth}
\caption{Comparison of {\em Poss-Flips} for GSAT, HSAT, NavRGSAT and
NavFGSAT.\label{tie-fig}}
\end{figure}
Figure~\ref{gsat_run} presents the dynamics of {\em poss-flips} during a
typical run of GSAT. It is easy to see that on early plateaux {\em possflips}
tend to grow with some random noise, for example, in
Figure~\ref{gsat_run}
second, third and fifth plateaux produced obvious growth of {\em possflips}
until drops corresponding to the improvement of the heuristic values and,
thus,
the end of the plateau. During the first and fourth plateaux, the growth
is not that
steady though still visible. Even though flips back are prohibited for
NRGSAT,
it maintains the same property, because of the high connectivity of the
problem
domain and the abundance of equally good choices.
Figure~\ref{gsat_tie} represents the average percentage of ties for a
3SAT problem
with 100 variables and 430 clauses over 100 runs for GSAT and HSAT, and
for GSAT
and NavRGSAT. The average number of {\em poss-flips} for GSAT dominates
the analogous
characteristic for HSAT by a noticeable amount. This type of dominance is
similar
in the comparison of GSAT with NavRGSAT in the beginning of the plateau
phase. In the
second part of the plateau phase the number of {\em poss-flips} for HSAT
or NavRGSAT
approaches the number of {\em poss-flips} for GSAT. Lower graph
represents second-order
ties for NavRGSAT that form a subset of {\em poss-flips}.
Our experiments confirmed the result obtained in \cite{gent:93} that the
whole picture
scales up linearly in the number of variables and the number of {\em
poss-flips}.
The plateau phase begins after about $0.2N-0.25N$ steps. By that moment
at most a quarter
of the variable set has been flipped, and thus NavFGSAT mimics HSAT up to
a certain degree.
After $2N$ or $3N$ flips, both versions of NavGSAT diverge significantly
from HSAT.
After these many steps both NavRGSAT and NavFGSAT still maintain random
oscillation
of {\em poss-flips}, whereas GSAT tends to promote higher levels of {\em
poss-flips}.
Unfortunately, for problems with larger ratio of the number of clauses to
the number
of variables NavFGSAT is often trapped in an infinite loop. NavRGSAT also
may behave
inefficiently for such problems: From time to time the policy of NavRGSAT
forces it to flip the same variable with a low weight several times in a
row to gain
the same weight as other variables from the set of {\em poss-flips}.
\putfig{gsat_run}{Dynamics of {\em Poss-Flips} for
GSAT.}{width=0.45\textwidth}
Thus, NavGSAT showed that the number of {\em poss-flips} plays an
important role
in improving the efficiency of GenSAT procedures. HSAT capitalizes on
this
property and therefore constitutes one of the most efficient hillclimbing procedures
for random problem instances. However, many real-world satisfiability
problems are
highly structured and, if applied, HSAT may easily fail due to its
queuing policy.
NavGSAT suggests another sub-family of GenSAT hill-climbing procedures
that does
not tend to increase the number of {\em poss-flips}. Weights of variables
and their
combinations can be used as a second tie-breaking heuristic to maintain
lower level
of {\em poss-flips} and find exact or deliver approximate solutions for
those
problems for which HSAT fails to solve.
\begin{figure}[t] \centering \
\psfig{figure=figs/gsat_and_hsat.eps,width=0.45\textwidth}
\psfig{figure=figs/gsat_and_navsat.eps,width=0.45\textwidth}
\caption{Percentage of {\em Poss-Flips} for GSAT with HSAT and GSAT with
NavRGSAT.\label{gsat_tie}}
\end{figure}
For randomly generated 3SAT problems HSAT proved to be one of the most
efficient
hill-climbing procedures. There has been reports on HSAT's failures in
solving
non-random propositional satisfiability problems \cite{gent:95}. We view
the
non-flexibility of HSAT's queue heuristic as a possible obstacle in
solving
over-constrained problems. This does not happen in solving random 3SAT
problems
with low $L/N$-ratio.
\section{Approximate Satisfaction}
While running experiments with GSAT, HSAT and other hill-climbing
procedures, we
noticed that GSAT experiences biggest loss in the performance in the
beginning of
the plateau phase where the amount of {\em poss-flips} can be as high as
20-25\%.
On the other hand, HSAT, NavFGSAT and NavRGSAT behave equally good during
the
hill-climbing phase and the beginning of the plateau phase.
We thus concluded that any of the latter three procedures can be applied
to
provide fast approximate solutions. For some problems, versions of
NavGSAT are
not as efficient as HSAT. Nonetheless, we introduced NavFGSAT and
NavRGSAT to
show that HSAT's queuing policy is not the unique way of improving the
efficiency of solving propositional satisfiability problems.
Approximate solutions can be utilized in time-critical problems where the
quality of the solution discounts the time spent for solving the problem.
NavGSAT can be also applied to problems with dynamically changing
domains, when the domain changes can influence the decision making
process.
Finally, approximate solution provide an excellent starting point for a
different search policy. For example, WGSAT and UGSAT \cite{frank:96}
utilized a promising idea of the instant heuristic update based on the
weight of unsatisfied clauses. An approximate solution provided by HSAT
or
NavGSAT constitutes an excellent starting point for WGSAT, UGSAT or
another
effective search procedure of a satisfiable solution, for example,
$\epsilon$-search (with heuristic updates). Among others we outline
the following benefits of employing HSAT or NavGSAT to deliver a good
starting point for another search method:
\begin{itemize}
\item Perfect initial assignment with a low number of unsatisfied
clauses.
\item Absence of hill-climbing phase that, for example, eliminates noise
in
tracking clause weights.
\item Efficient search in both steps of policy-switching approach.
\item Convenient point in time to fork search in multi-agent/multiprocessor
problem scenarios.
\end{itemize}
Although HSAT, itself, is an efficient hill-climbing procedure for
randomly
generated problems with a low clause-variable ratio, we expect that HSAT
might experience difficulties in more constrained problems. NavGSAT
provides
another heuristic that guides efficiently in the initial phases.
On the other hand, the hill-climbing phase may either produce noise in
clause
weight bookkeeping or redundant list of vertices with updated heuristics
that slows down the performance of $\epsilon$-search. Search with
policy switching can benefit significantly from employing efficient
procedures in all of its phases.
\section{Navigational Problems}
To confirm the results of our navigational approach to GSAT, we applied
all the
discussed above hill-climbing procedures to the following simple
navigational
problem:
\begin{quote}
{\bf Navigational Problem (NavP):}\ An agent is given a task to find the
shortest path
that reaches a goal vertex from a starting vertex in an ``obstacle-free''
rectangular grid.
\end{quote}
NavP is a simplistic planning problem. It can be represented as a
propositional satisfiability problem with $N=|S|*D$ variables, where $S$
is the
set of vertices in the rectangular grid and $D$ is the shortest distance
between
starting vertex $X$ and goal vertex $G$. In a correct solution, a
variable
$x^d_s$ is assigned {\em True} ($x^d_s = 1$), if $s$ is $d$th vertex on
the
shortest path from $X$ to $G$, and {\em False} otherwise. There can be
only one
variable with the {\em True} value among variables representing grid
vertices
that are $d$-far from starting vertex $X$. This requirement implies
$L_1 =\Theta(|S|^2*D)$ pigeonhole-like constraints:
\[ \bigwedge_{d=1}^{D-1} \bigwedge_{s_1\not = s_2} (\neg V^d_{s_1} \vee
\neg V^d_{s_2}) \]
Already these constraints make the domain look ``over-constrained,''
since
the ratio of $L_1$ to $N$ is not asymptotically bounded. Another group of
constraints has to force {\em True}-valued variables to form a
continuous path.
There can be different ways of presenting such constraints, we chose the
easiest and the most natural presentation that does not produce extra
variables:
\[ \bigwedge_{d=2}^{D-1} \bigvee_{s\in S} (V^d_s\wedge (V^{d-1}_{s_1}
\vee
V^{d-1}_{s_2} \vee V^{d-1}_{s_3} \vee V^{d-1}_{s_4})) \]
Vertices $s_1, s_2, s_3, s_4\in S$ are the neighbors of vertex $s\in S$
in the rectangular
grid. To reduce the number of variables and clauses, the initial and the
goal
states are represented by stand-alone single clauses:
\[ (V^1_{s_5} \vee V^1_{s_6} \vee V^1_{s_7} \vee V^1_{s_8}) \]
\[ (V^{D-1}_{s_9} \vee V^{D-1}_{s_{10}} \vee V^{D-1}_{s_{11}} \vee V^{D1}_{s_{12}}) \]
Vertices $s_5, s_6, s_7, s_8\in S$ are the neighbors of the starting
vertex,
$s_9, s_{10}, s_{11}, s_{12}\in S$ are the neighbors of the goal vertex.
Second group of constraints is not presented in the classical CNF form.
It is possible to reduce it to 3SAT, but such a reduction will introduce
a
lot of new variables and clauses and will significantly slow down the
performance
without facilitating search for a satisfiable assignment. From the point
of view
of hill-climbing procedures that track clause weights, this would mean
only
a different initial weight assignment and a linear change in bookkeeping
clause weights. Therefore, we decided to stay with the original non-3SAT
model
and considered each complex conjunction $\bigvee_{s\in S} (V^d_s\wedge
(V^{d-1}_{s_1} \vee
V^{d-1}_{s_2} \vee V^{d-1}_{s_3} \vee V^{d-1}_{s_4}))$ as a single
clause.
Together with the starting and goal vertex constraints, the second group
contains $L_2 =\Theta(D)$ constraints that force {\em True}-valued
variables to form a continuous path.
It is fairly easy to come up with an initial solution, so that all but
one constraint
are satisfied. Figure~\ref{NavP} shows one of such solutions that
alternates
between the goal vertex and one of its neighbors, and the final path that
satisfies all the constraints.
The original GSAT has complexity that is exponential on $D$. It performs
poorly for such domains, because at every step it has more equally good
chances
than any other algorithm.
%\footnote{That was how we got the idea that the amount of ties is
important for
%random problem instances too.}.
HSAT was able to solve ``toy'' problems with less than 200 variables
until its search was under the influence
of initial states. For larger problems, after an initial search HSAT used
to switch to a systematic
search that avoided changing recently changed vertices. Since HSAT restarted search
from both starting and goal vertices on a regular basis, all the variable
corresponding
to their neighboring vertices has frequently changed their values.
Therefore, paths
from the opposite direction attempted to avoid changing these variables
again.
This was one of the domain where the queuing policy of HSAT played
against it.
A slightly modified versions of NavFGSAT and NavRGSAT were capable of
solving
larger problems using Top-Down Depth-First-Search (TDDFS). TDDFS
traverses
repeatedly the search tree (the set of vertices reachable in $D$ steps)
from the
root down, each time attempting to visit the least visited vertex from
the current
vertex or, if possible, unvisited vertex. The only modification of this
behavior was
that NavGSAT methods alternated roots between the starting vertex and the
goal
vertex while performing such search.
\begin{figure}[t] \centering \
\psfig{figure=../slides/Shortest_Path.ps,width=0.4\textwidth}
\psfig{figure=../slides/Shortest_path.ps,width=0.4\textwidth}
\caption{Initial and final solutions for NavP.\label{NavP}}
\end{figure}
\section{Conclusions}
We showed that GenSAT hill-climbing procedures for solving propositional
satisfiability problems can be interpreted as navigational, agentcentered search
on a high-dimensional cube, NavGSAT. This type of search heavily depends
on how well
heuristic values represent the actual distance towards the set of goal
states.
HSAT, one of the most efficient GSAT-like procedures, maintains low level
of
{\em poss-flips}. We identified this property as the main benefit of HSAT
in
comparison with the original GSAT. However, the non-flexibility of HSAT's
queuing policy can be an obstacle in solving more constrained problems.
We introduced two versions of NavGSAT that also maintain low level of
{\em poss-flips} and can be applied as approximating procedures for
time-critical or dynamically changing problems, or serve as a starting
phase in search procedures with switching search policies.
\section*{Acknowledgements}
This research is sponsored in part by the National Science Foundation
under grant number IRI-9502548. The views and conclusions contained in
this
document are those of the authors and should not be interpreted as
representing
the official policies, either expressed or implied, of the sponsoring
organizations or the U.S. government.
\begin{thebibliography}{EPIA-97}
\end{document}
GSAT is a local hill-climbing procedure for solving propositional
satisfiability problems. We re-consider it as a search procedure
performed
on an $N$-dimensional cube by a fictitious agent with a limited
lookahead.
In this paper we analyze a few variations of GSAT in the following
directions:
we apply recently developed theory of agent-centered search to the
field of GSAT problems, derive fast approximating procedures based on
variable weights that can be efficiently combined with other search
procedures
or utilized as the initial phase of policy-switching search algorithms,
and argue about the reasons of higher effectiveness of HSAT compared with
one of the original GSAT.